It's simplest to define the mapping by through:
[tex]x=r\cos\theta, y=r\sin\theta[/tex]

This sets up a bijection almost everywhere between (x,y) and [tex](r,\theta)[/tex].
(That is with (x,y) on the plane, r on the non-negative half-axis, and [tex]\theta[/tex] on the half-open interval [tex][0,2\pi)[/tex]

the theta involved in this equation is not the theta of polar/cylindrical coordinates (i.e. constrained in [0, 2pi)), is it?

I had succeeded in proving this equation but it involved treating the polar angle has being free to take any value in [itex](-\infty, \infty)[/itex]. I was trying to justify that it was justified to do that. But now I'm a little confused. Is it justified?